# Spin quantum number

The **spin quantum number** is the quantum number, written s, that fixes the magnitude of a particle's intrinsic angular momentum, or spin. For a given type of particle only one value of s is allowed: electrons, protons, and neutrons have s = 1/2, photons have s = 1, and pions have s = 0.<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup> The allowed values run through the integers and half-integers, s = 0, 1/2, 1, 3/2, and so on, and this single number divides every particle into two families, bosons and fermions, with sharply different collective behavior.<sup>[2](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m11/spin.html)</sup><sup> • </sup><sup>[3](https://quantumatlas.umd.edu/entry/spin/)</sup>

| Key fact | Detail |
|---|---|
| Definition | s is defined as the quantum number of intrinsic spin angular momentum, quantized independently of orbital angular momentum<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup> |
| Allowed values | s = 0, 1/2, 1, 3/2, ... ; one fixed value per particle type<sup>[2](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m11/spin.html)</sup><sup> • </sup><sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup> |
| Magnitude formula | \( S = \sqrt{s(s+1)}\,\hbar \), so an electron (s = 1/2) has \( S^2 = \tfrac{3}{4}\hbar^2 \)<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup><sup> • </sup><sup>[2](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m11/spin.html)</sup> |
| Projection quantum number | \( S_z = m_s \hbar \) with \( m_s = -s, -s+1, \ldots, +s \); for electrons \( m_s = \pm 1/2 \) (spin up, spin down)<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup><sup> • </sup><sup>[2](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m11/spin.html)</sup> |
| Spin-statistics link | Integer-spin particles are bosons; half-integer-spin particles are fermions<sup>[4](https://farside.ph.utexas.edu/teaching/qm/Quantum/node95.html)</sup> |
| Experimental basis | The 1921 Stern-Gerlach experiment showed silver atoms deflected into two orientations, evidence of spin quantization<sup>[5](https://hyperphysics.phy-astr.gsu.edu/hbase/spin.html)</sup><sup> • </sup><sup>[6](https://qm.utasphys.cloud.edu.au/01-stern-gerlach/1-1-exp/)</sup> |
| Practical use | Proton spin alignment in hydrogen under a strong magnetic field is the basis of magnetic resonance imaging (MRI)<sup>[3](https://quantumatlas.umd.edu/entry/spin/)</sup> |

## Magnitude and projection

Spin angular momentum is quantized in the same mathematical form as orbital angular momentum. The magnitude of the spin of a particle with spin quantum number s is

\[ S = \sqrt{s(s+1)}\,\hbar, \]

where \( \hbar = h/2\pi \) is the reduced [Planck constant](https://www.edgepedia.org/planck-constant). For an electron, s = 1/2, so \( S^2 = s(s+1)\hbar^2 = \tfrac{3}{4}\hbar^2 \).<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup><sup> • </sup><sup>[2](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m11/spin.html)</sup> This parallels the orbital formula \( L = \sqrt{l(l+1)}\,\hbar \), except that l can take many values within an atom while s is fixed for a given particle.<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup>

The direction of spin is also quantized. Along any chosen axis, conventionally the z-axis defined by a magnetic field, the component is

\[ S_z = m_s \hbar, \qquad m_s = -s,\ -s+1,\ \ldots,\ +s, \]

so a particle of spin s has \( 2s+1 \) possible projections. For an electron these are \( m_s = +1/2 \), called spin up, and \( m_s = -1/2 \), called spin down.<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup> A measurement of an electron's spin along a magnetic field therefore yields one of two results and never an intermediate angle.<sup>[3](https://quantumatlas.umd.edu/entry/spin/)</sup> Measurements along different axes are incompatible: passing a spin-up-along-z beam into an apparatus oriented along x again yields two outcomes with equal probability.<sup>[7](https://sites.esm.psu.edu/~vfm5153/IQM/chapter1.html)</sup>

## Intrinsic, not classical rotation

Spin is a purely quantum-mechanical property with no classical analog. It is not associated with any rotating internal parts of the particle; an electron has spin even though it is believed to be a point particle with no internal structure.<sup>[2](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m11/spin.html)</sup> The analogy to a planet spinning on its axis, often used in introductions, is only rough.<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup>

Spin is also independent of the other quantum numbers of a bound electron. In an atom the state is labeled by \( (n, l, m_l, m_s) \), the principal, orbital, orbital-projection, and spin-projection quantum numbers, and the electron's spin is always s = 1/2 regardless of n, l, and \( m_l \).<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup>

## Experimental evidence: Stern-Gerlach

In 1921, [Otto Stern](https://www.edgepedia.org/otto-stern) and Walter Gerlach performed the experiment that revealed spin quantization directly. A beam of silver atoms was sent through a non-uniform magnetic field gradient; the force on each atom's magnetic moment, \( F_z \approx \mu_z \, \partial B_z / \partial z \), deflected the beam in proportion to \( S_z \).<sup>[5](https://hyperphysics.phy-astr.gsu.edu/hbase/spin.html)</sup><sup> • </sup><sup>[6](https://qm.utasphys.cloud.edu.au/01-stern-gerlach/1-1-exp/)</sup>

**Two spots, not a smear.** [Classical physics](https://www.edgepedia.org/classical-physics) predicts a continuous range of spin projections between \( -|\mathbf{S}| \) and \( +|\mathbf{S}| \) for a thermal beam, and so a continuous smear on the detector. Instead, only two projections were observed, consistent with \( S_z = \pm\hbar/2 \).<sup>[6](https://qm.utasphys.cloud.edu.au/01-stern-gerlach/1-1-exp/)</sup> The total angular momentum of a silver atom is half-valued, traceable to its unpaired electron's spin, which is why the beam splits into exactly two parts.<sup>[8](https://batelaan.unl.edu/electron-stern-gerlach-magnets/)</sup>

## The spin-statistics connection

The spin quantum number does more than label a particle; it determines the particle's quantum statistics. The spin-statistics theorem states that particles of integer spin (0, \( \hbar \), \( 2\hbar \), ...) are bosons, obeying Bose-Einstein statistics, while particles of half-integer spin (\( \hbar/2 \), \( 3\hbar/2 \), ...) are fermions, obeying Fermi-Dirac statistics.<sup>[4](https://farside.ph.utexas.edu/teaching/qm/Quantum/node95.html)</sup><sup> • </sup><sup>[9](https://homes.lorentz.leidenuniv.nl/~pjhdent/SecQuant08.pdf)</sup>

In non-relativistic quantum mechanics this rule must be accepted as an empirical fact. In relativistic quantum field theory it can be proved, though the proof is complicated; in [Steven Weinberg](https://www.edgepedia.org/steven-weinberg)'s formulation, causality, Poincaré invariance, positive energies, and positive probability together imply the spin-statistics relation.<sup>[4](https://farside.ph.utexas.edu/teaching/qm/Quantum/node95.html)</sup><sup> • </sup><sup>[9](https://homes.lorentz.leidenuniv.nl/~pjhdent/SecQuant08.pdf)</sup><sup> • </sup><sup>[10](https://www.phy.olemiss.edu/~luca/Topics/qft/spin-stat.html)</sup>

**Consequences of the two families.** Any number of bosons can occupy the same quantum state, which is why photons in a laser beam pass through one another freely. Fermions obey the [Pauli exclusion principle](https://www.edgepedia.org/pauli-exclusion-principle), which allows at most one particle per quantum state; for electrons this means no two electrons in a multi-electron system can possess identical sets of observable eigenvalues.<sup>[3](https://quantumatlas.umd.edu/entry/spin/)</sup><sup> • </sup><sup>[11](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/08%3A_Quantum_Statistical_Mechanics/8.01%3A_Prelude_to_Quantum_Statistical_Mechanics)</sup><sup> • </sup><sup>[4](https://farside.ph.utexas.edu/teaching/qm/Quantum/node95.html)</sup> The exclusion principle is the reason for the structure of the periodic table of elements.<sup>[3](https://quantumatlas.umd.edu/entry/spin/)</sup>

## Values for common particles

Electrons, protons, and neutrons are all spin-1/2 particles and hence fermions, the basic components of matter. Photons are spin-1 bosons that convey forces; the photon's spin is responsible for the polarization of light. Pions have s = 0.<sup>[1](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)</sup><sup> • </sup><sup>[2](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m11/spin.html)</sup><sup> • </sup><sup>[3](https://quantumatlas.umd.edu/entry/spin/)</sup> Composite particles such as nuclei and atoms get their spin by adding the spin vectors of their constituents, so composites can be either integer or half-integer spin.<sup>[2](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m11/spin.html)</sup>

The half-integer/integer distinction has practical reach beyond classification. In magnetic resonance imaging, the proton spins of hydrogen in the body align with a strong applied magnetic field; radio-frequency pulses alter the spins, and the signal released as they relax back builds up the image.<sup>[3](https://quantumatlas.umd.edu/entry/spin/)</sup>

## References

1. [Quantum Numbers and Rules, College Physics 2e, OpenStax](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)
2. [Spin, Physics 222 module, University of Tennessee](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m11/spin.html)
3. [Spin, The Quantum Atlas, University of Maryland](https://quantumatlas.umd.edu/entry/spin/)
4. [Spin Statistics Theorem, University of Texas quantum mechanics course notes](https://farside.ph.utexas.edu/teaching/qm/Quantum/node95.html)
5. [Electron Spin, HyperPhysics, Georgia State University](https://hyperphysics.phy-astr.gsu.edu/hbase/spin.html)
6. [The Stern-Gerlach experiment, Quantum Mechanics course, University of Tasmania](https://qm.utasphys.cloud.edu.au/01-stern-gerlach/1-1-exp/)
7. [Chapter 1: Stern-Gerlach Experiments, Introduction to Quantum Mechanics, Penn State](https://sites.esm.psu.edu/~vfm5153/IQM/chapter1.html)
8. [Electron Stern-Gerlach Magnets, Herman Batelaan, University of Nebraska](https://batelaan.unl.edu/electron-stern-gerlach-magnets/)
9. [The spin-statistics theorem, second quantization lecture notes, Leiden University](https://homes.lorentz.leidenuniv.nl/~pjhdent/SecQuant08.pdf)
10. [Spin-Statistics Theorem, Topics in theoretical physics, University of Mississippi](https://www.phy.olemiss.edu/~luca/Topics/qft/spin-stat.html)
11. [Prelude to Quantum Statistical Mechanics, Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/08%3A_Quantum_Statistical_Mechanics/8.01%3A_Prelude_to_Quantum_Statistical_Mechanics)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Quantum states and wave functions › Quantum numbers › Spin quantum numbers and spin states*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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